Unweighted Index Numbers

There are two methods of constructing unweighted index numbers: (1) Simple Aggregative Method (2) Simple Average of Relative Method

Simple Aggregative Method

In this method, the total price of commodities in a given (current) year is divided by the total price of commodities in a base year and expressed as percentage:

{P_{on}} = \frac{{\sum {P_n}}}{{\sum {P_o}}} \times 100

Simple Average of Relative Method

In this method, we compute price relatives or link relatives of the given commodities and then use one of the averages such as the arithmetic mean, geometric mean, median, etc. If we use the arithmetic mean as the average, then:

{P_{on}} = \frac{1}{n}\sum \left( {\frac{{{P_n}}}{{{P_o}}}} \right) \times 100

The simple average of relative method is simpler and easier to apply than the simple aggregative method. The only disadvantage is that it gives equal weight to all items.

Example:

The following are the prices of four different commodities for 1990 and1991. Compute a price index with the (1) simple aggregative method and (2) average of price relative method by using both the arithmetic mean and geometric mean, taking 1990 as the base.

Commodity
Cotton
Wheat
Rice
Grams
Price in 1990
909
288
767
659
Price in 1991
874
305
910
573

 

Solution:
The necessary calculations are given below:

Commodity
Price in 1990
{P_o}
Price in 1991
{P_n}
Price Relative
P = \frac{{{P_n}}}{{{P_o}}} \times 100
\log P
Cotton
909
874
\frac{{874}}{{909}} \times 100 = 69.15
1.9829
Wheat
288
305
\frac{{305}}{{288}} \times 100 = 105.90
2.0249
Rice
767
910
\frac{{910}}{{767}} \times 100 = 118.64
2.0742
Grams
659
573
\frac{{573}}{{659}} \times 100 = 86.95
1.9393
Total
\sum {P_o} = 2623
\sum {P_n} = 2662
\sum P = 407.64
\begin{gathered} \sum \log P \\ = 8.0213\\ \end{gathered}

(1) Simple Aggregative Method:

{P_{on}} = \frac{{\sum {P_n}}}{{\sum {P_o}}} \times 100 = \frac{{2662}}{{2623}} \times 100 = 101.49

(2) Average of Price Relative Method (using the arithmetic mean):

{P_{on}} = \frac{1}{n}\sum \left( {\frac{{{P_n}}}{{{P_o}}}} \right) \times 100 = \frac{1}{4}\left( {407.64} \right) = 101.91

Average of Price Relative Method (using the geometric mean):

{P_{on}} = anti\log \left( {\frac{{\sum \log P}}{n}} \right) = anti\log \left( {\frac{{8.0213}}{4}} \right) = 101.23